Riemann problem

Summary

A Riemann problem, named after Bernhard Riemann, is a specific initial value problem composed of a conservation equation together with piecewise constant initial data which has a single discontinuity in the domain of interest. The Riemann problem is very useful for the understanding of equations like Euler conservation equations because all properties, such as shocks and rarefaction waves, appear as characteristics in the solution. It also gives an exact solution to some complex nonlinear equations, such as the Euler equations. In numerical analysis, Riemann problems appear in a natural way in finite volume methods for the solution of conservation law equations due to the discreteness of the grid. For that it is widely used in computational fluid dynamics and in computational magnetohydrodynamics simulations. In these fields, Riemann problems are calculated using Riemann solvers. The Riemann problem in linearized gas dynamics As a simple example, we investigate the properti

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On the energy dissipation rate of solutions to the compressible isentropic Euler system

Elisabetta Chiodaroli

In this paper we extend and complement the results in [4] on the well-posedness issue for weak solutions of the compressible isentropic Euler system in 2 space dimensions with pressure law

p(\rho) = \rho^\gamme, \geq 1

. First we show that every Riemann problem whose one dimensional self-similar solution consists of two shocks admits also in_nitely many two dimensional admissible bounded weak solutions (not containing vacuum) generated by the method of De Lellis and

Sz\’{e}kelyhidi

[11], [12]. Moreover we prove that for some of these Riemann problems and for

1\leq < 3

such solutions have greater energy dissipation rate than the self-similar solution emanating from the same Riemann data. We therefore show that the maximal dissipation criterion proposed by Dafermos in [7] does not favour the classical self similar solutions.

Springer Verlag2014

This overview is concerned with the well-posedness problem for the isentropic compressible Euler equations of gas dynamics. The results we present are in line with the programof investigatingthe efficiency of different selection criteria proposed in the literature in order to weed out non-physical solutions to more-dimensional systems of conservation laws and they build upon the method of convex integration developed by De Lellis and Sz,kelyhidi for the incompressible Euler equations. Mainly following [5], we investigate the role of the maximal dissipation criterion proposed by Dafermos in [6]: we prove how, for specific pressure laws, some non-standard (i.e. constructed via convex integration methods) solutions to the Riemann problem for the isentropic Euler system in two space dimensions have greater energy dissipation rate than the classical self-similar solution emanating from the same Riemann data. We therefore show that the maximal dissipation criterion proposed by Dafermos does not favour in general the self-similar solutions.

Springer Heidelberg2016

Non-uniqueness of admissible weak solutions to the Riemann problem for isentropic Euler equations

Elisabetta Chiodaroli

We study the Riemann problem for multidimensional compressible isentropic Euler equations. Using the framework developed in Chiodaroli et al (2015 Commun. Pure Appl. Math. 68 1157-90), and based on the techniques of De Lellis and Szekelyhidi (2010 Arch. Ration. Mech. Anal. 195 225-60), we extend the results of Chiodaroli and Kreml (2014 Arch. Ration. Mech. Anal. 214 1019-49) and prove that it is possible to characterize a set of Riemann data, giving rise to a self-similar solution consisting of one admissible shock and one rarefaction wave, for which the problem also admits infinitely many admissible weak solutions.

IOP PUBLISHING LTD2018

On the energy dissipation rate of solutions to the compressible isentropic Euler system

Elisabetta Chiodaroli

In this paper we extend and complement the results in [4] on the well-posedness issue for weak solutions of the compressible isentropic Euler system in 2 space dimensions with pressure law

p(\rho) = \rho^\gamme, \geq 1

Sz\’{e}kelyhidi

[11], [12]. Moreover we prove that for some of these Riemann problems and for

1\leq < 3

Springer Verlag2014